What did reformers see as the major challenges facing American society in the 1820s and 1803s and how did they respond to them?
I would say that what most of the reformers from this era had in common was that they felt that American society was being threatened by the change toward a more market-based economy. In past times, most Americans had not produced things to sell to others -- they had been self-sufficient, with only a very few items obtained by buying them or trading for them.
As America moved more toward a market economy, the old social ties started to break down. There came to be more opportunities for individuals to leave their old ways of life and, for example, go to the city. As the old ways broke down, people started to think that more and more change was possible.
Reformers responded to these changes in one of two ways -- they tried to turn the clock back to a time when things were more community-based (think Shakers or the LDS) or they whole-heartedly embraced the idea of change and tried to make what they saw as a new, more perfect world (the women's movement, abolitionism).
Reformers in the 1820s and 1830s began to see that the changing face of America provided its own sense of unique challenges which had to be faced in some form. The emergence of slave states vs. free states began to mobilize forces on both sides in terms of the abolitionist and anti- abolitionist movement. The issue of slavery would be the fundamental cause of the Civil War in thirty years' time, so it began to be the driving force behind reform at the time. Another movement which began during this time period and would have profound impacts on how American society saw itself would be the emergence of the American literary tradition, specifically Poe, Cooper, and Emerson. The seeds of transcendentalism began to be laid in this time period with its emphasis on nature, emotion over reason, and the changing perception of how individuals see themselves. These ideas would occupy critical importance over time and serve as essential catalysts in the causing of the Civil War.
There was also a large scale movement to reform society at this time. These reformers felt that the promise of a democratic society through the Constitution had gone unfulfilled by the early 1800's, and they wanted to improve society to live up to the promise.
Reformers such as Dorothea Dix (Prisons and Mental Hospitals), William Lloyd Garrison (Abolition), Horace Mann (Education) and Henry David Thoreau (Transcendentalism) and the women of the early temperence movement represented the generation born after the Revolution and Constitution took place. Their reform efforts, while mostly unsuccessful, also laid the groundwork for the Progressives of the later 19th century.